Poisson Brackets for some Coulomb Branches

TL;DR

This paper constructs explicit Poisson brackets for operators generating the Coulomb branch in certain 3d N=4 quiver gauge theories, revealing algebraic structures of these moduli spaces.

Contribution

It provides explicit Poisson bracket relations for Coulomb branch operators in specific theories, using geometric and symmetry-based methods, and conjectures brackets for related Higgs branches.

Findings

Explicit Poisson brackets for ADE and minimal A2 Coulomb branches.

Methodology combining geometric properties and symmetry charges.

Conjectured brackets for Higgs branches from 6d and 5d theories.

Abstract

We construct Poisson bracket relations between the operators which generate the chiral ring of the Coulomb branch of certain $3d$ $\mathcal{N}=4$ quiver gauge theories. In the case where the Coulomb branch is a free space, $ADE$ Klein singularity, or the minimal $A_2$ nilpotent orbit, we explicitly compute the Poisson brackets between the generators using either inherited properties of the abstract Coulomb branch variety, or the expected charges of these operators under the global symmetry (known through use of the monopole formula). We also conjecture Poisson brackets for Higgs branches that originate from $6d$ theories with tensionless strings or $5d$ theories with massless instantons for which the HWG is known, based on representation theoretic and operator content constraints known from the study of their magnetic quiver.